Orthonormalization, polar decomposition and transformation to an effective hamiltonian.

What is it about?

(Uploaded: 2018-04-18). Prior to Bloch (1958), the theory for transforming a perturbed Hamilto-nian H = H0+V to an effective Hamiltonian Heff was thought of as a problem of constructing uni-tary matrices which diminishes certain parts of the matrix for H. Bloch instead started geometri-cally: First he projected the exact perturbed eigenfunctions "down" on the zeroth order space in question. Then he defined the "wave operator" as the inverse mapping "up" to the space of the exact. His Heff constructed from these two maps is not hermitian, but the wave operator obeys a simple non-linear equation from which its formal expansion, together with that of Hess, can be determined very easily. Wishing a hermitian Heff with orthonormal eigenvectors, des Cloizeaux (and later Soliverez) constructed a hermitian and positive operator that orthonormalizes Bloch's projected eigenvectors of H. To formulate these adjustments, des Cloizeaux used Bloch's idea of writing the various operators as sums of ket-bra terms. He was not aware of Löwdin's orthonormalization methods, but it was actually the symmetric he re-invented and described in general terms before applying it to the perturbation problem. It is carried it out by acting on the given vectors with the inverse square root of a very simple positive (and hence hermitian) operator B. To find the needed power of this B, he introduced an orthonormal set its eigenvectors - which turn out to be those obtained by applying Löwdin's canonical orthonormalization to the given vectors instead of the symmetric. I take des Cloizeaux' operator B as an outset for a reconsideration of Löwdin's symmetric and canonical methods - keeping a special focus on geometry and on the problem of transforming to an effective Hamiltonian.

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Why is it important?

After Löwdin's introduction and use of his orthonormalization methods, other authors have found additional valuable properties and uses. It is an aim of the present work is to collect and develop some of these in efficient ways which reveal new connections. Starting as described above with des Cloizeaux's operator B, we immediately understand how the canonical orthonormalized vectors come about. This B is (trivially) is invariant under a permutation of the given vectors - a fact which lead directly to the Slater-Coster Theorem from 1954. And leads to a strong help for decomposition into irreducible subspaces for a group. Furthermore: It turns out that symmetric orthonormalization together with the Carlson-Keller Theorem from 1957 are very closely related to polar decomposition - a method I consider as a generalization to operators of the fact any complex number z can be written as r*u where r is the numerical value of z and u is the number on the unit circle which is nearest to z. In this generalization, distance between operators is measured by the Euclidian norm. Klein used the symmetric orthonormalization to uniquely characterize des Cloizeaux's transformation to a hermitian Heff as the one which affects the exact eigenvectors of H minimally. Now I characterize it as the one which is nearest to the Bloch's projection operator.

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